# Decay rates at infinity for solutions to periodic Schr\"{o}dinger equations

## TL;DR

This paper investigates decay properties of solutions to periodic Schrödinger equations in exterior domains, proving the absence of superexponentially decaying solutions and providing a new proof of the spectrum's absolute continuity.

## Contribution

It establishes decay rate limitations for solutions of periodic Schrödinger equations and offers a novel proof for the spectrum's absolute continuity in such operators.

## Key findings

- Superexponentially decaying solutions do not exist for these equations.
- The spectrum of certain periodic Schrödinger operators is absolutely continuous.
- Provides a new approach to spectral analysis of periodic elliptic operators.

## Abstract

We consider the equation $\Delta u=Vu$ in exterior domains in $\mathbb{R}^2$ and $\mathbb{R}^3$, where $V$ has certain periodicity properties. In particular we show that such equations cannot have non-trivial superexponentially decaying solutions. As an application this leads to a new proof for the absolute continuity of the spectrum of particular periodic Schr\"{o}dinger operators. The equation $\Delta u=Vu$ is studied as part of a broader class of elliptic evolution equations.